The manifest success of the Finite Element Method has led to progressively increased demands being made of it. In particular, there is increasing pressure to use sophisticated 3D models which result in large costly numerical systems. It has shown that the singular point of the fundamental solutions can validly be taken outside the domain of the problem thereby yielding regular boundary integral equations (1]. The location of the singular point outside the domain of the problem permits the use of harmonic functions with higher order singularities, as kernels in the boundary integral equations. This possibility is attractive because the higher order singularities decay rapidly away from the singular point thereby resulting in more diagonally dominant algebraic equations. In this paper, a set of higher order functions are used as kernel functions, which still satisfy the governing equation (Lapalce's equation) with discre-tization acheived as continuous, discontinuous and partially discontinuous elements.